# A Complex Network Theory-Based Modeling Framework for Unmanned Aerial Vehicle Swarms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. UAV Swarming System

#### 2.1. System Description

- (a)
- Vehicle: The swarming formation consists of vehicles (individual UAVs). Based on the report “Sustaining America’s Precision Strike Advantage” issued by the U.S. Center for Strategic and Budgetary Assessment (CSBA) in 2015, small UAVs will be the main formation vehicle used to consume enemy weapons [31]. Thus, current research has focused on small and low-cost UAVs, such as the DARPA “elf” UAV, the “Coyote” UAV of the LOCUST project, and the “Partridge” UAVs of the U.S. Navy [32,33]. These vehicles are better for swarming owing to their small size, low cost, and test repeatability, among other attributes.
- (b)
- Payload: Payload refers to equipment and sensors related to UAV missions. Sensors, radars, camera equipment, and weapons are the most common payloads [34]. In a typical UAV swarming system, the individual UAV limits the variety of payloads it can carry for technical reasons, especially for small UAVs; thus, the payloads are generally integrated with the aircraft [33,35]. As a result, a UAV swarming system may contain UAVs with different payloads when performing missions. For example, for cooperative detection, the system may be equipped with heterogeneous sensors.
- (c)
- Datalink: The communication datalink is the basis for the realization of UAV swarm control and the successful implementation of missions. Two kinds of datalinks are commonly used: a traditional datalink and a UAV ad hoc network [36]. The traditional datalink can be further divided into “ground station to UAV” and “ground station to satellite to UAV” links [37]. A communication ad hoc network is the network formed by multiple UAVs. It will be the main communication mode in the future or the next generation of communication datalinks. At present, three kinds of communication ad hoc networks (mobile ad hoc networks, wireless sensor networks, and wireless mesh networks) can be utilized in a UAV swarm owing to their mobility and network topology dynamics, multiple-hop transmission, and self-organization [37,38,39]. Moreover, these networks are rather robust, so that a single-node failure in the network has no effect on the performance of the entire network.

#### 2.2. Scope Identification

## 3. Modeling Framework

#### 3.1. Network Representation Based on Complex Network

_{a}, G

_{b}, and G

_{c}, respectively. Then, to represent the network topology, three networks are abstracted into graphs based on graph theory.

_{i}(i = 1, 2, …, m) aircraft, where n

_{1}+ n

_{2}+ …+ n

_{m}= n. Then, the three network layers and their interlayer relationships can be described as follows:

_{a}= (V

_{ai}, E

_{aj}, W

_{aj}), where V

_{ai}is the set of nodes, V

_{ai}= {V

_{a}

_{1}, V

_{a}

_{2}…, V

_{an}}; and E

_{aj}is the set of edges, E

_{aj}= {E

_{a}

_{1}, E

_{a}

_{2}, …, E

_{aN}

_{1}} (j = 1, 2, 3, …, N

_{1}, N

_{1}> n). Moreover, the communication network keeps changing, which will affect the communication quality. Thus, the communication layer is defined as a weighted network, with the weight denoted W

_{aj}, where W

_{aj}= {W

_{a}

_{1}, W

_{a}

_{2}…, W

_{aN}

_{1}}.

_{b}can be defined as G

_{b}= (V

_{bi}, E

_{bj}, W

_{bj}), where V

_{bi}is the set of nodes, V

_{bi}= {V

_{b}

_{1}, V

_{b}

_{2}, …, V

_{bn}}; and E

_{bj}is the set of edges, E

_{bj}= {E

_{b}

_{1}, E

_{b}

_{2}, …, E

_{bN}

_{2}} (j = 1, 2, 3, …, N

_{2}, N

_{2}> n). The weight is denoted W

_{bj}= {W

_{b}

_{1}, W

_{b}

_{2}, …, W

_{bN}

_{2}}.

_{c}. Thus, G

_{c}can be written as G

_{c}= (V

_{ci}, E

_{ci}, W

_{c}), where V

_{ci}is the set of nodes, V

_{ci}= {V

_{b}

_{1}, V

_{b}

_{2}, …, V

_{cn}}; and E

_{cj}is the set of edges, E

_{cj}= {E

_{c}

_{1}, E

_{c}

_{2}, …, E

_{cN}

_{3}} (j = 1, 2, 3, …, N

_{3}, N

_{3}> n).

_{i}(i = 1, 2, 3, …, n) and L(b − c)

_{i}are defined as the interlayer relationships between the communication layer and structure layer and between the structure layer and mission layer, respectively. The dynamic relationships are depicted in Figure 3 and Figure 4 which is the removal of nodes in one layer will lead to the removal of nodes in other layers and also the removal of interlayer edges.

#### 3.2. Modeling Algorithm

**a. Modeling Algorithm for Structure 1**

- Initialization: Generate n nodes and define the number of payload types, m; the number of nodes under each payload, n
_{i}(i = 1,2, …, m); and the number of hierarchy levels, p. - Connection:
- (a)
- Adds edges between nodes and adjacent nodes in the communication layer and structural layer according to topology;
- (b)
- Adds edges between every two nodes among n
_{i}nodes; - (c)
- Adds edges one-to-one between the communication layer and structure layer and between the structure layer and the mission layer separately;
- (d)
- Multiple edges and self-loops should not exist.

- Weight: Randomly assign a weight to the edges in the communication layer and structure layer based on mission requirements. The weight of edges in the mission layer should assign the same value.
- Output: After all of the nodes, edges, and weights are generated, output the network.

**b. Modeling Algorithm for Structure 2**

_{j}. In this paper, the number of leaders is set equal to the number of payload types to simplify the modeling difficulty. Then, one can build the network via the following steps (the pseudocode is shown in Table 2):

- Initialization: Generate n nodes and define the number of payload types, m, and the number of nodes under each payload, n
_{i}(i = 1,2, …, m). - Connection:
- (a)
- Adds edges between leader nodes and follower nodes in the communication layer and structural layer;
- (b)
- Adds edges between every follower node and its adjacent nodes;
- (c)
- Adds edges between every two nodes among n
_{i}nodes; - (d)
- Adds edges one-to-one between the communication layer and structure layer and between the structure layer and the mission layer separately;
- (e)
- Multiple edges and self-loops should not exist.

- Weight: Randomly assign a weight to the edges in the communication layer and structure layer based on mission requirements. The weight of edges in the mission layer should be assigned the same value.
- Output: After all the nodes, edges, and weights are generated, output the network.

**c. Modeling Algorithm for Structure 3**

- Initialization: Generate n nodes and define the number of payload types, m, and the number of nodes under each payload, n
_{i}(i = 1, 2, …, m). - Connection:
- (a)
- Adds edges among nodes and random [k − 1,k + 1] nodes in the communication layer and the structural layer;
- (b)
- Adds edges between every two nodes among n
_{i}nodes; - (c)
- Adds edges one-to-one between the communication layer and the structure layer and between the structure layer and the mission layer separately;
- (d)
- Multiple edges and self-loop should not exist.

- Weight: Randomly assign a weight to the edges in the communication layer and the structure layer based on mission requirements. The weight of edges in the mission layer should be assigned the same value.
- Output: After all of the nodes, edges, and weights are generated, output the network.

#### 3.3. Network Measurements

_{qi}calculated from its network layer, which can be denoted k

_{qi}(G

_{q}) (q = a,b,c), and (b) the degree calculated by the connections with nodes in another layer, which can be denoted ${k}_{qi}^{l}$, and the following conditions must be satisfied:

_{qi}in the UAV swarming network can be expressed as:

_{qi}among nodes V

_{qi}and its k

_{qi}adjacent nodes and the total possible edges k

_{qi}(k

_{qi}− 1)⁄2, i.e.:

_{qi}(G

_{q}) is denoted the connections in the same layer while ${E}_{qi}^{l}$ is the connection with nodes in different layers, and the following conditions must be satisfied:

## 4. Case Study Analysis and Discussion

#### 4.1. Case Study

#### 4.2. Topology Analysis and Discussion

#### 4.2.1. Analysis of Degree and Degree Distribution of Nodes

^{2}greater than 0.80. So, the UAV swarming networks have the topological properties of the scale-free network. One can know from the characteristic of the scale-free network that this network may be highly resistant to random attacks and be quite sensitive to targeted attacks [24].

#### 4.2.2. Analysis of Average Shortest Path Length

#### 4.2.3. Analysis of Clustering Coefficient

#### 4.2.4. Analysis of Small-World Characteristics

#### 4.2.5. Dynamic Topology Analysis

#### 4.3. Robustness Evaluation

_{degree}and t

_{path}are the average degree of nodes and average length path, respectively.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 21.**Comparison of average shortest path length between multi-layer network and single network.

**Figure 22.**Comparison of average clustering coefficient between multi-layer network and single network.

Algorithm 1: Modeling Algorithm for the control structure of Behavior-based methods |

1: Initialization: V_{a}[1…n], V_{b}[1…n], V_{c}[1…n], n, m, n[1…m],2: for i←1 to n do3: addedge&weight [(V_{a}[i], V_{b}[i], weight), (V_{b}[i], V_{c}[i], weight)]4: for x←1 to p do5: for j←1 to x do6: addedge&weight[(V_{a}[i], V_{a}[i+x], weight), (V_{b}[i], V_{b}[i+x], weight)]7: addedge&weight [(V_{a}[i], V_{a}[i+x+1], weight), (V_{b}[i], V_{b}[i+x+1], weight)]8: addedge&weight [(V_{a}[i+x], V_{a}[i+x+1], weight), (V_{b}[i+x], V_{b}[i+x+1], weight)]9: end for10: end for11: end for12: for i←1 to m do13: for j←1 to n[i] do14: for x←j to n[i] do15: addedge&weight (V_{c}[j], V_{c}[j+1], weight)16: end for17: end for18: end for19: return G(G_{a}, G_{b}, G_{c}) |

Algorithm 2: Modeling Algorithm for the Control Structure of Leader-Follower Strategy |

1: Initialization: V_{a}[1…n], V_{b}[1…n], V_{c}[1…n], n, m, n[1…m]2: for i←1 to n do3: addedge&weight [(V_{a}[i], V_{b}[i], weight), (V_{b}[i], V_{c}[i], weight)]4: end for5: for i←1 to m − 1 do6: addedge&weight (V_{a}[i], V_{a}[i+1], weight)7: end for8: for i←1 to m do9: for j←1 to n[i]-1 do10: addedge&weight [(V_{a}[i], V_{a}[j], weight), (V_{b}[i], V_{b}[j], weight)]11: end for12: for x←j to n[i] do13: addedge&weight (V_{c}[j], V_{c}[j+1], weight)14: end for15: for j←1 to n[i]-3 do16: addedge&weight [(V_{a}[j], V_{a}[j+1], weight), (V_{b}[j], V_{b}[j+1], weight)]17: addedge&weight [(V_{a}[j], V_{a}[j+2], weight), (V_{b}[j], V_{b}[j+2], weight)]18: end for19: end for20: return G(G_{a}, G_{b}, G_{c}) |

Algorithm 3: Modeling Algorithm for the Autonomous Control Structure |

1: Initialization: V_{a}[1…n], V_{b}[1…n], V_{c}[1…n], n, m, n[1…m], k2: for i←1 to n do3: addedge&weight [(V_{a}[i], V_{b}[i], weight), (V_{b}[i], V_{c}[i], weight)]4: end for5: for i←1 to k do6: for j←1 to m do7: randomly choose nodes V_{a}, V_{b}8: if V_{a}, V_{b} ≠ V_{a}[j], V_{b}[j] then9: addedge&weight [(V_{a}, V_{a}[j], weight),(V_{b}, V_{b}[j], weight)]10: end if11: end for12: end for13: for i←1 to m do14: for j←1 to n[i]-1 do15: for x←j to n[i] do16: addedge&weight (V_{c}[j], V_{c}[j+1], weight)17: end for18: end for19: end for20: return G(G_{a}, G_{b}, G_{c}) |

Features | Structure 1 | Structure 2 | Structure 3 |
---|---|---|---|

Number of nodes | 165 | 165 | 165 |

Number of edges | 655 | 655 | 695 |

Average degree | 7.94 | 7.94 | 8.42 |

Cluster coefficient | 0.44 | 0.52 | 0.30 |

Average length path | 4.46 | 4.28 | 3.14 |

Features | Random Network 1 | Random Network 2 | Random Network 3 |
---|---|---|---|

Number of nodes | 165 | 165 | 165 |

Number of edges | 655 | 655 | 695 |

Cluster coefficient | 0.03 | 0.03 | 0.04 |

Average length path | 2.68 | 2.68 | 2.60 |

Features | Structure 1 | Structure 2 | Structure 3 |
---|---|---|---|

Average degree of nodes | −0.06022 | −0.06027 | −0.06023 |

Average length path | −0.04863 | −0.04866 | −0.04886 |

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**MDPI and ACS Style**

Wang, L.; Lu, D.; Zhang, Y.; Wang, X.
A Complex Network Theory-Based Modeling Framework for Unmanned Aerial Vehicle Swarms. *Sensors* **2018**, *18*, 3434.
https://doi.org/10.3390/s18103434

**AMA Style**

Wang L, Lu D, Zhang Y, Wang X.
A Complex Network Theory-Based Modeling Framework for Unmanned Aerial Vehicle Swarms. *Sensors*. 2018; 18(10):3434.
https://doi.org/10.3390/s18103434

**Chicago/Turabian Style**

Wang, Lizhi, Dawei Lu, Yuan Zhang, and Xiaohong Wang.
2018. "A Complex Network Theory-Based Modeling Framework for Unmanned Aerial Vehicle Swarms" *Sensors* 18, no. 10: 3434.
https://doi.org/10.3390/s18103434